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In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables.

In first-order logic with identity with constant symbols $a$ and $b$ , the sentence $Q(a)\lor P(b)$ is a ground formula. A ground expression is a ground term or ground formula.

Examples

Consider the following expressions in first order logic over a signature containing the constant symbols $0$ and $1$ for the numbers 0 and 1, respectively, a unary function symbol $s$ for the successor function and a binary function symbol $+$ for addition.

$s(0),s(s(0)),s(s(s(0))),\ldots$ are ground terms;
$0+1,\;0+1+1,\ldots$ are ground terms;
$0+s(0),\;s(0)+s(0),\;s(0)+s(s(0))+0$ are ground terms;
$x+s(1)$ and $s(x)$ are terms, but not ground terms;
$s(0)=1$ and $0+0=0$ are ground formulae.

Formal definitions

What follows is a formal definition for first-order languages. Let a first-order language be given, with $C$ the set of constant symbols, $F$ the set of functional operators, and $P$ the set of predicate symbols.

Ground term

A ground term is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):

Elements of $C$ are ground terms;
If $f\in F$ is an $n$ -ary function symbol and $\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}$ are ground terms, then $f\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)$ is a ground term.
Every ground term can be given by a finite application of the above two rules (there are no other ground terms; in particular, predicates cannot be ground terms).

Roughly speaking, the Herbrand universe is the set of all ground terms.

Ground atom

A ground predicate, ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.

If $p\in P$ is an $n$ -ary predicate symbol and $\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}$ are ground terms, then $p\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)$ is a ground predicate or ground atom.

Roughly speaking, the Herbrand base is the set of all ground atoms,^[1] while a Herbrand interpretation assigns a truth value to each ground atom in the base.

Ground formula

A ground formula or ground clause is a formula without variables.

Ground formulas may be defined by syntactic recursion as follows:

A ground atom is a ground formula.
If $\varphi$ and $\psi$ are ground formulas, then $\lnot \varphi$ , $\varphi \lor \psi$ , and $\varphi \land \psi$ are ground formulas.

Ground formulas are a particular kind of closed formulas.

References

^ Alex Sakharov. "Ground Atom". MathWorld. Retrieved October 20, 2022.

Dalal, M. (2000), "Logic-based computer programming paradigms", in Rosen, K.H.; Michaels, J.G. (eds.), Handbook of discrete and combinatorial mathematics, p. 68
Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6
First-Order Logic: Syntax and Semantics

[1] Alex Sakharov. "Ground Atom". MathWorld. Retrieved October 20, 2022.

[1]

@@ Line 1: / Line 1: @@
 {{Short description|Term that does not contain any variables}}
+{{Formal languages}}
 In [[mathematical logic]], a '''ground term''' of a [[formal system]] is a [[Term (logic)|term]] that does not contain any [[Variable (mathematics)|variables]]. Similarly, a '''ground formula''' is a [[Well formed formula|formula]] that does not contain any variables.
-In [[First-order logic#Equality and its axioms|first-order logic with identity]], the [[Sentence (mathematical logic)|sentence]] <math>Q(a) \lor P(b)</math> is a ground formula, with <math>a</math> and <math>b</math> being constant symbols. A '''ground expression''' is a ground term or ground formula.
+In [[First-order logic#Equality and its axioms|first-order logic with identity]] with constant symbols <math>a</math> and <math>b</math>, the [[Sentence (mathematical logic)|sentence]] <math>Q(a) \lor P(b)</math> is a ground formula. A '''ground expression''' is a ground term or ground formula.
 ==Examples==
-Consider the following expressions in [[first order logic]] over a [[signature (mathematical logic)|signature]] containing a constant symbol <math>0</math> for the number <math>0,</math> a unary function symbol <math>s</math> for the successor function and a binary function symbol <math>+</math> for addition.
+Consider the following expressions in [[first order logic]] over a [[signature (mathematical logic)|signature]] containing the constant symbols <math>0</math> and <math>1</math> for the numbers 0 and 1, respectively, a unary function symbol <math>s</math> for the successor function and a binary function symbol <math>+</math> for addition.
-* <math>s(0), s(s(0)), s(s(s(0))), \ldots</math> are ground terms,
+* <math>s(0), s(s(0)), s(s(s(0))), \ldots</math> are ground terms;
-* <math>0 + 1, \; 0 + 1 + 1, \ldots</math> are ground terms,
+* <math>0 + 1, \; 0 + 1 + 1, \ldots</math> are ground terms;
-* <math>x + s(1)</math> and <math>s(x)</math> are terms, but not ground terms,
+* <math>0+s(0), \; s(0)+ s(0), \; s(0)+s(s(0))+0</math> are ground terms;
-* <math>s(0) = 1</math> and <math>0 + 0 = 0</math> are ground formulae,
+* <math>x + s(1)</math> and <math>s(x)</math> are terms, but not ground terms;
+* <math>s(0) = 1</math> and <math>0 + 0 = 0</math> are ground formulae.
-==Formal definition==
+==Formal definitions==
-What follows is a formal definition for [[first-order language]]s. Let a first-order language be given, with <math>C</math> the set of constant symbols, <math>V</math> the set of (individual) variables, <math>F</math> the set of functional operators, and <math>P</math> the set of [[predicate symbol]]s.
+What follows is a formal definition for [[first-order language]]s. Let a first-order language be given, with <math>C</math> the set of constant symbols, <math>F</math> the set of functional operators, and <math>P</math> the set of [[predicate symbol]]s.
-===Ground terms===
+===Ground term===
-A '''{{visible anchor|ground term}}''' is a [[Term (logic)|terms]] that contain no variables. They may be defined by logical recursion (formula-recursion):
+A '''{{visible anchor|ground term}}''' is a [[Term (logic)|term]] that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):
 # Elements of <math>C</math> are ground terms;
 # If <math>f \in F</math> is an <math>n</math>-ary function symbol and <math>\alpha_1, \alpha_2, \ldots, \alpha_n</math> are ground terms, then <math>f\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)</math> is a ground term.
@@ Line 31: / Line 33: @@
 If <math>p \in P</math> is an <math>n</math>-ary predicate symbol and <math>\alpha_1, \alpha_2, \ldots, \alpha_n</math> are ground terms, then <math>p\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)</math> is a ground predicate or ground atom.
-Roughly speaking, the [[Herbrand base]] is the set of all ground atoms, while a [[Herbrand interpretation]] assigns a [[truth value]] to each ground atom in the base.
+Roughly speaking, the [[Herbrand base]] is the set of all ground atoms,<ref>{{MathWorld |id=GroundAtom |title=Ground Atom |author=Alex Sakharov |access-date=October 20, 2022 |ref= }}</ref> while a [[Herbrand interpretation]] assigns a [[truth value]] to each ground atom in the base.
 ===Ground formula===
@@ Line 37: / Line 39: @@
 A '''{{visible anchor|ground formula}}''' or '''{{visible anchor|ground clause}}''' is a formula without variables.
-Formulas with free variables may be defined by syntactic recursion as follows:
+Ground formulas may be defined by syntactic recursion as follows:
+# A ground atom is a ground formula.
-# The free variables of an unground atom are all variables occurring in it.
-# The free variables of <math>\lnot p</math> are the same as those of <math>p.</math> The free variables of <math>p \lor q, p \land q, p \to q</math> are those free variables of <math>p</math> or free variables of <math>q.</math>
+# If <math>\varphi</math> and <math>\psi</math> are ground formulas, then <math>\lnot \varphi</math>, <math>\varphi \lor \psi</math>, and <math>\varphi \land \psi</math> are ground formulas.
-# The free variables of <math>\forall x \; p</math> and <math>\exists x \; p</math> are the free variables of <math>p</math> except <math>x.</math>
+Ground formulas are a particular kind of [[Sentence (mathematical logic)|closed formulas]].
 ==See also==